Abstract: The goal of this minisymposium is to bring together researchers work-
ing on problems related to the nonlocal modeling of physical phenomena
and their mathematical analysis. The theme is on modeling, analysis and
simulation with a focus on nonlocal continuum equations that arise from
applications. The session will be multifaceted so as to cover work related
nonlocal modeling and computational simulations of models, and analyti-
cal and numerical aspects such as well-posedness of nonlocal stationary and
evolution equations, regularity of solutions and numerical approximations.

Nonlocal mathematical models arise naturally in many important fields
and they are found to be useful where classical (local) models cease to be
predictive. Moreover, nonlocal models are suitable for multiscale modeling
as they can be effective in capturing the underlying nonsmooth microscale
fields. An example is peridynamics, a nonlocal reformulation of the basic
equations of motion of continuum mechanics, which is being used to model
cracks and discontinuous fields in solid mechanics. Other areas of application
include image processing, modeling population aggregation, wave propaga-
tion, pattern formation, and porous media flow. In this minisymposium,
research works which have produced novel analytical and numerical methods
for nonlocal problems will be presented.

MS-We-D-09-113:30--14:00Localization of nonlocal gradients and some applications in variational convergenceMengesha, Tadele (The Univ. of Tennessee)Abstract: We study weighted directed difference quotients and their localization to classical notions of derivatives in several function spaces. We will characterize vector fields in the space of Sobolev spaces, space of BV functions and space of BD functions in a unified way. As an application, we will use the characterization and localization mechanism to compute Gamma limits of some nonlocal functionals that appear in peridynamics.

MS-We-D-09-214:00--14:30A fast numerical method for nonlocal modelsWang, Hong (Univ. of South Carolina)Abstract: Peridynamic/nonlocal diffusion models provide a very effective modeling of phenomena with long range interactions and nonlocal behavior. However, these models involve complex and singular integral operators. Consequently, resulting numerical methods generate dense matrices, for which direct solvers require O(N^3) computational complexity and O(N^2) memory for a problem of size N. This imposes significant computational and memory challenge in realistic applications.
We present a fast numerical method for a nonlocal diffusion model by exploiting the structure

MS-We-D-09-314:30--15:00A Nonlocal Strain Measure for Digital Image CorrelationLehoucq, Richard (Sandia National Labs)Abstract: We propose a nonlocal strain measure for use with digital image correlation (DIC). Whereas the traditional notion of compatibility (strain as the derivative of the displacement field) is problematic when the displacement field varies substantially either because of measurement noise or material irregularity, the proposed measure remains robust, well-defined and invariant under rigid body motion. Moreover, when the displacement field is smooth, the classical and nonlocal strain are in agreement.

MS-We-D-09-415:00--15:30Local boundary conditions in nonlocal problemsCeliker, Fatih (Wayne State Univ.)Abstract: We study nonlocal wave equations on bounded domains related to peridynamics. We display a methodology for enforcing boundary conditions (periodic, antiperiodic, Dirichlet, or Neumann) through an integral convolution. We present a numerical study of the approximate solution, study convergence order with respect to the polynomial order of approximation, and observe optimal convergence. We depict solutions for each boundary condition to ascertain the behavior of waves under the nonlocal theory.